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class. 'A mathematician is one to whom that is as obvious as that
twice two makes four is to you. Liouville was a mathematician.*
Young Hermite's pioneering work in Abelian functions, well
begun before he was twenty-one, was as far beyond Kelvin's
example in unobviousness as the example is beyond 'twice two
makes four.' Remembering the cordial welcome the aged
Legendre had accorded the revolutionary work of the young
and unknown Jacobi, Liouville guessed that Jacobi would show
a similar generosity to the beginning Hermite. He was not
The first of Hermite1 s astonishing letters to Jacobi is dated
from Paris, January 1843. 'The study of your [Jacobf s] memoir
on quadruple periodic functions arising in the theory of Abelian
functions has led me to a theorem, for the division of the argu-
ments [variables] of these functions, analogous to that which
you gave ... to obtain the simplest expression for the roots of
the equations treated by Abel. M. Liouville induced me to
write to you, to submit this work to you; dare I hope, Sir, that
you will be pleased to welcome it with all the indulgence it
needs?' With that he plunges at once into the mathematics.
To recall briefly the bare nature of the problem in question:
the trigonometric functions are functions of one variable with
one period, thus sin (x + %TT) = sin #, where x is the variable
and 2?r is the period; Abel and Jacobi, by 'inverting* the elliptic
integrals, had discovered functions of one variable and feoo
periods, say f(x -j- p -f q) = /(#), where p, q are the periods (see
Chapters 12, 18); Jacobi had discovered functions of fceo
variables and four periods, say
F(x + a + b, y + c + d) = F(xsy),
where ajbtc,d are the periods. A problem early encountered hi
trigonometry is to express sin (- ), or sin { -}, or generally sin
W        W
( -), where n is any given integer, in terms of sin a; (and
possibly other trigonometric functions of or). The correspond-
ing problem for the functions of two variables and four periods
was that which Hermite attacked. In the trigonometric pro-